530 Magnetic Measurements /28 : 3 



where F is the proportionality constant 



F-IH*A» (4) 



In practice, F is determined from Equation 3 by finding/ for a sample 

 of known susceptibility k in the instrument. 



Having found the susceptibility k in emu, it is possible to compute 

 the number of free electrons per molecule. If the molar concentration 

 is c (the moles per milliliter is 10 ~ 3 c), one may define a molar suscepti- 

 bility Xm by 



10 3 

 Xm = — x 



It can be shown that x m f° r many paramagnetic atoms is given by 



N* g p 2 J(J + i) 



Xm = 



3kT 



where iVis Avogadro's number, g the Lande factor, 2 /3 the Bohr magneton, 

 J the total momentum quantum number, k the Boltzmann gas constant, 

 and T the absolute temperature. For a paramagnetic molecule (instead 

 of a single atom) , J is replaced by S, the spin quantum number, and g by 

 the spin-only value, g s . For a single electron, g s = 2. 



The theoretical basis of the Rankine balance is the same as that of the 

 Gouy balance. However, instead of having a stationary magnet and ob- 

 serving the force on the sample, it uses a fixed sample holder and measures 

 the force on a suspended magnet. A diagram illustrating this principle 

 is shown in Figure 2. The quartz suspending fibers tend to minimize 

 the effects of the earth's magnetic field, and the symmetry of the sus- 

 pension reduces the effects of vibrations. The mirror is part of an 

 optical lever system for measuring the rotational displacements of the 

 suspension accompanying changes in force upon the magnet. 



In the Johnson Foundation instrument, two samples are used, one on 

 each side of the magnet. These are in the form of half cylinders. The 

 difference in susceptibility between the solutions in the two half cylinders 



2 The Lande g factor is the ratio 



p Imc 

 g ~ ]x"~e~ 



where p is the total angular moment, /jl the magnetic moment, c the velocity in a 

 vacuum, and e the charge on an electron. The Lande g factor is always between 

 one and two. In terms of quantum numbers 



. J(J + 1) + S(S + 1) - L(L + 1) 

 g = 1 + 



2 J (J + 1) 



